Search for long lived heaviest nuclei beyond the valley of stability

PHYSICAL REVIEW C 77, 044603 (2008)

Search for long lived heaviest nuclei beyond the valley of stability

P. Roy Chowdhury,1,* C. Samanta,1,2,3,† and D. N. Basu4,‡1Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700 064, India

2Physics Department, Virginia Commonwealth University, Richmond, Virginia 23284-2000, USA3Physics Department, University of Richmond, Richmond, Virginia 23173, USA4Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700 064, India

(Received 13 August 2007; revised manuscript received 9 January 2008; published 23 April 2008)

The existence of long lived superheavy nuclei (SHN) is controlled mainly by spontaneous fission and α-decayprocesses. According to microscopic nuclear theory, spherical shell effects at Z = 114, 120, 126 and N = 184provide the extra stability to such SHN to have long enough lifetime to be observed. To investigate whetherthe so-called “stability island” could really exist around the above Z, N values, the α-decay half-lives alongwith the spontaneous fission and β-decay half-lives of such nuclei are studied. The α-decay half-lives of SHNwith Z = 102–120 are calculated in a quantum tunneling model with DDM3Y effective nuclear interactionusing Qα values from three different mass formulas prescribed by Koura-Uno-Tachibana-Yamada (KUTY),Myers-Swiatecki (MS), and Muntian-Hofmann-Patyk-Sobiczewski (MMM). Calculation of spontaneous fission(SF) half-lives for the same SHN are carried out using a phenomenological formula and compared with SFhalf-lives predicted by Smolanczuk et al. A possible source of discrepancy between the calculated α-decayhalf-lives of some nuclei and the experimental data of GSI, JINR-FLNR, RIKEN, is discussed. In the regionof Z = 106–108 with N ∼ 160–164, the β-stable SHN 268

106Sg162 is predicted to have highest α-decay half-life (Tα ∼ 3.2 h) using Qα value from MMM. Interestingly, it is much greater than the recently measuredTα (∼22 s) of deformed doubly magic 270

108Hs162 nucleus. A few fission-survived long-lived SHN which are eitherβ-stable or having large β-decay half-lives are predicted to exist near 294110184, 293110183, 296112184, and 298114184.These nuclei might decay predominantly through α-particle emission.

DOI: 10.1103/PhysRevC.77.044603 PACS number(s): 27.90.+b, 23.60.+e, 21.10.Hw, 21.30.Fe

I. INTRODUCTION

The theoretical studies of properties of heaviest nucleiduring the past few decades have drawn considerable at-tention of experimentalists to investigate the existence ofsuperheavy nuclei beyond the valley of stability. Since themacroscopic [1,2] description on the basis of liquid dropmodel (LDM) does not take the shell effect into account, itfails to explain the variation of fission barrier height of heavynuclei with the increase of the fissility parameter (∼Z2/A).However, according to modern nuclear theory, hindrance tothe fissioning of heavy nuclei would be enhanced due to thepresence of deformed and spherical shell closures. Differentsemimicroscopic approaches, e.g., macroscopic-microscopicmodel (MMM) [3–6] and its modification [7] include pairingand nuclear shell effects [8] to reproduce the properties ofground and deformed states of nuclei. Many purely mi-croscopic [9–13] descriptions like Hartree-Fock-Bogoliubov(HFB) model with zero range forces of Skyrme type [14,15] orfinite range forces of Gogny type [16,17] and relativistic meanfield (RMF) [18,19] theory predict the possible deformed andspherical neutron shell closures at N = 162 and N = 184 [20],respectively. Since strong influence of nuclear shells [21] in theregion of superheavy elements might make sufficiently longlived SHN to be observed, the search for heavier elements inthe natural samples was started [22–25] about 30 years ago.

*[email protected]†[email protected]; [email protected]‡[email protected]

Experimental investigations in finding the SHN around Z =107–118 have been pursued mainly at three different places:Gesellschaft fur Schwerionenforschung (GSI) in Darmstadt(Germany), Joint Institute for Nuclear Research (JINR) inDubna (Russia), and RIKEN, Japan. In the beginning of the1980’s the first observations of the elements with Z = 107–109 were made at GSI [26]. In 1994, α-decay chains wereobserved from nucleus 269110 [27] and later on, α-decay chainsfrom nuclides 271110, 272111, 277112 [28–30], 283112 [31] weredetected at GSI.

While RIKEN claimed discovery of the 278113 SHN[32,33], it also reconfirmed the α decay chains from 271110[34], 272111 [35], and 277112 [36]. Observations of the α de-cay chains of nuclei 294118,290−293116,288,287115,286−289114,282−284113, 285,283112 [37], 278−280111, 273,281110 [38,39],274−276109, 275108, 272,270107, 271106 were reported byJINR [40–43].

Recently, the 270Hs (Z = 108, N = 162) SHN has beenproduced in the 26Mg+248Cm reaction [44]. According tothe theoretical calculations [3,45], nucleus 270Hs162 (Z = 108)should have the features of “deformed doubly magic” nucleus.Most of the heaviest nuclei are expected to be deformed dueto partial filling of large nuclear shells by outer nucleons.Dvorak et al. measured the energy (Eα) of α particle emittedfrom 270Hs162 and used the value of Eα to calculate Qα (9.02 ±0.03 MeV) for the α decay of 270Hs162. A phenomenologicalformula [46] estimated the α decay half-life (∼22 s).

Earlier, it was believed [23,47–49] that traditional sphericalsuperheavy nuclei might form an “island of stability” centeredaround 298114184 separated from the “peninsula” of known

0556-2813/2008/77(4)/044603(10) 044603-1 ©2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevC.77.044603

P. ROY CHOWDHURY, C. SAMANTA, AND D. N. BASU PHYSICAL REVIEW C 77, 044603 (2008)

nuclei by a region of deep instability. Due to both deformedneutron shell and proton shell effects at Z = 108 and N = 162the extension of the peninsula of known nuclei might connectthe stability island of spherical superheavy nuclei arounddoubly magic spherical Z = 114 proton shell and sphericalN = 184 neutron shell. Since fission barrier and shell effectplay very important role for the existence of long livedsuperheavy nuclei it is crucial to determine the fission barrierand half-life of fissioning nucleus with a good accuracy. Itis well known that very small barrier height against fissioncan break the nucleus into two fragments immediately afterit is formed. The α decay of superheavy nuclei [50–56] ispossible if the shell effect supplies the extra binding energy andincreases the barrier height of fission. β-stable nuclei havingrelatively longer half-life for spontaneous fission than that forα decay indicates that dominant decay mode for such SHNmight be α-decay.

In our previous works [51,54,57–59] we showed theapplicability of the microscopic calculation in predicting theα decay half-lives of SHN from a direct comparison withthe experimental data [40–42]. However, as a number of SHNwere predicted to have relatively large α decay half-lives, itis necessary to find out whether those SHN would survivethe fission [60] and β-decay. In such cases those nuclei can bedetected in the laboratory through α decay. This work exploresthe possibility of finding long lived SHN by comparing thecalculated α decay half-lives (Tα) with available theoreticalspontaneous fission (SF) half-lives [61,62], calculated β-decayhalf-lives (Tβ) [63] and the experimental data on SF. Theα decay half-lives of SHN with Z = 102–120 are calculated ina quantum tunneling model with DDM3Y effective nuclear in-teraction using Qα values from three different mass formulas.

A brief outline of the methodology of the present calcula-tion is presented in Sec. II. Spontaneous fission half-lives fromboth phenomenological [64,65] and microscopic approach[61,62] are given in Sec. III. Finally, in Sec. IV, resultsand discussions and in Sec. V, summary and conclusion arepresented.

II. FORMALISM

The α decay half-lives are calculated in the framework ofquantum mechanical tunneling of an α particle from a parentnucleus [57] and the results are shown in Figs. 1–3. The detailsof calculation of the α decay half-lives of superheavy nucleiwere described in our earlier works [57–59]. The required nu-clear interaction potentials are calculated by double folding thedensity distribution functions of the α particle and the daughternucleus with density dependent M3Y effective interaction. Themicroscopic α-nucleus potential thus obtained, along with theCoulomb interaction potential and the minimum centrifugalbarrier required for the spin-parity conservation, form thepotential barrier. The spin-parity conservation condition in adecay process is fulfilled if and only if

J = J1 + J2 + l, π = π1.π2.(−1)l , (1)

where J, J1, and J2 are the spins of the parent, daughter, andemitted nuclei, respectively, π , π1, and π2 are the parities

of the parent, daughter, and emitted nuclei, respectively,and l is the orbital angular momentum carried away inthe process. This conservation law, thus, forces a minimumangular momentum to be carried away in the decay process.Consequently, contribution of the angular momentum givesrise to a centrifugal barrier

Vl = h̄2l(l + 1)/(2µR2), (2)

where µ is the reduced mass of the daughter and emitted nucleisystem and R is the distance between them.

The half-lives of α disintegration processes are calculatedusing the WKB approximation for barrier penetrability. Spher-ical charge distributions have been used for calculating theCoulomb interaction potentials. Most of the experimentalQ-values of α decay (Qα) are obtained from experiments donein GSI, Germany and JINR, Dubna. For theoretical Q-values,mass formulas from KUTY [66], Myers-Swiatecki [67], andMuntian et al. [7,68,69] are used.

The experimental decay Q values (Qex) have been obtainedfrom the measured α particle kinetic energies Eα using thefollowing expression:

Qex =(

Ap

Ap − 4

)Eα + (

65.3Z7/5p − 80.0Z2/5

p

)10−6 MeV,

(3)

where the first term is the standard recoil correction and thesecond term is an electron shielding correction in a systematic

FIG. 1. (Color online) Plots of α decay half-life [log10(T1/2/sec)]in logarithmic scale versus proton number Z for different massnumber A (indicated on top of each coloumn) using zero angularmomentum transfer (� = 0). (a) bar coded columns are theoreticalhalf lives (Tα M3Y Q-MS) in WKB frame work with DDM3Yinteraction and [QMS

th ] from Myers-Swiatecki mass formula, (b)columns filled with dots (Tα M3Y Q-K) are in the same framework butwith [QKUTY

th ] from Koura-Tachibana-Uno-Yamada mass estimates,(c) solid columns are theoretical half lives (Tα M3Y Q-M) in WKBframe work with DDM3Y interaction and [QM

th ] from Muntian-Patyk-Hofmann-Sobiczewski mass formula, (d) hollow columns areexperimental α decay half lives (Tα Expt). Experimental errors aregiven in Table I.

044603-2

SEARCH FOR LONG LIVED HEAVIEST NUCLEI BEYOND . . . PHYSICAL REVIEW C 77, 044603 (2008)

Z=102

N120 140 160 180 200

T1/

2(s

ec)

10-1210-910-610-310010310610910121015101810211024102710301033103610391042

Z=104

N

N N

120 140 160 180 200

T1/

2(s

ec)

10-12

10-9

10-6

10-3

100

103

106

109

1012

1015

1018

1021

1024

1027

1030

Z=106

120 140 160 180 200

T1/

2(s

ec)

10-1210-1010-810-610-410-21001021041061081010101210141016101810201022 Tα M3Y Q-K

Tα M3Y Q-M

Tα SM

Tα Expt

Tsf SM

Tsf Expt

Tβ

Z=108

140 160 180 200

T1/

2(s

ec)

10-12

10-10

10-8

10-6

10-4

10-2

100

102

104

106

108

1010

1012

1014

(a) (b)

(c) (d)

FIG. 2. (Color online) Variation of α de-cay and fission half-lives with neutron num-ber for elements (a) Z = 102, (b) Z = 104,(c) Z = 106, (d) Z = 108 are shown. For allplots the following symbols are used: Dash-dotted line (Tα M3Y Q-K) and continuous linewith square symbols (Tα M3Y Q-M) representα decay half-lives calculation using Q-valuesfrom KUTY (Q-K) and Muntian et al. (Q-M) respectively in this work. Triangle sym-bol (Tα SM) represents α-decay half-lives pre-dicted within a microscopic framework [61,62]. Hexagon symbol (Tα Expt) representsmeasured α decay half-lives. Solid circle(Tsf SM) represents fission half-lives predictedby microscopic calculation. Diamond symbol(Tsf expt) represents measured fission half-lives for some nuclei. Line-inverted triangle(Tβ ) shows β decay half-lives predicted inRef. [63].

manner as suggested by Perlman and Rasmussen [70]. Ap andZp are mass number and atomic number of parent nucleus,respectively.

The theoretical decay Q values Qth have been obtainedfrom theoretical estimates for the atomic mass excesses [7,66–69] using the following relationship:

Qth = M − (Mα + Md ) = �M − (�Mα + �Md ), (4)

which if positive allows the decay, where M , Mα , Md and�M , �Mα , �Md are the atomic masses and the atomic massexcesses of the parent nucleus, the emitted α particle, and theresidual daughter nucleus, respectively, all expressed in theunits of energy. As Qα-value appears inside the exponentialintegral as well as in denominator of the expression [57,58] ofα decay half-lives, the entire calculation is very sensitive tothe Q-values.

III. PHENOMENOLOGICAL AND MICROSCOPICCALCULATIONS FOR SPONTANEOUS FISSION

HALF-LIVES

Spontaneous fission of heavy nuclei was first observedby Flerov and Petrjak in 1940 [71] from the 238U nucleus.

Spontaneous fission and α decay are the main decay modes[72–74] of superheavy nuclei. β-decay could be anotherpossible decay mode for the superheavy nuclei lying beyondthe β-stability line of the nuclear chart. However, sincethe β-decay proceeds via weak interaction, the process isslow and less favored and energy involved (released) is alsoless compared to spontaneous fission and α decay whichproceed quickly via strong interaction making these processesmore probable. For heaviest nuclei, the mutual repulsion ofelectric charge is higher than surface energy of the nucleusarising from the short range nuclear forces. On the basisof liquid drop model, Bohr and Wheeler [75] describedthe mechanism of nuclear fission and established a limitfor Z2/A ∼ 48 for spontaneous fission. Beyond this limitnuclei are unstable against spontaneous fission. Accordingto this fact, no nucleus beyond Z ∼ 100 can exist due tovery small fission barrier. Both theoretical and experimentalinvestigations on superheavy nuclei (SHN) support the factthat a bound superheavy nucleus can be formed only due toshell effects.

A simple semi-empirical formula on spontaneous fissionhalf-lives for even-even, odd A and odd-odd nuclei in theground state was proposed by Swiatecki in 1955 [76]. Byincluding the deviation of experimental ground state massesfrom a smooth reference surface based on liquid drop model,

044603-3


Z=110

T1/

2(s

ec)

10-1210-1010-810-610-410-210010210410610810101012101410161018

Z=112

10-1210-1010-810-610-410-210010210410610810101012101410161018

Z=114

T1/

2(s

ec)

10-1510-1310-1110-910-710-510-310-1101103105107109101110131015

Z=116

10-1510-1310-1110-910-710-510-310-1101103105107109101110131015

(b)

(c) (d)

Z=118

N140 160 180 200

T1/

2(s

ec)

10-6

10-4

10-2

100

102

104

106

108

Z=120

N140 160 180 200

10-6

10-4

10-2

100

102

104

106

108(e) (f)

(a)

Tα M3Y Q-K

Tα M3Y Q-M

Tα SM

Tα Expt

Tsf SM

Tsf Expt

Tβ

FIG. 3. (Color online) Same as Fig. 2 forelements (a) Z = 110, (b) Z = 112, (c) Z = 114,(d) Z = 116, (e) Z = 118, (f) Z = 120.

Swiatecki successfully reproduced the experimental data withhis semi-empirical formula [76]. Microscopic calculation ofspontaneous fission half-lives is very difficult due to boththe complexity of the fission process and the uncertainty ofthe height and shape of the fission barrier [77]. Ren andXu [64,65] generalized the formulas of spontaneous fissionhalf-lives of even-even nuclei in their ground state to both,the case of odd nuclei and the case of fission isomers [65].The spontaneous fission half-lives of odd-A nuclei and ofodd-odd nuclei in the ground state were calculated by usingthe generalized form of the Swiatecki’s formula and by a newformula where the blocking effect of unpaired nucleon on thehalf-lives were taken into account. By introducing a blockingfactor or a generalized seniority in the formulas of the half-livesof even-even nuclei, the experimental fission half-lives ofodd-A nuclei and of odd-odd nuclei with Z = 90 to Z = 108were reasonably reproduced with the same parameters used inground state of even-even nuclei.

In the present work, spontaneous fission half-lives for bothneutron deficient and neutron rich isotopes of elements Z =102–120 have been calculated using the following formula

[65]:

log10(T1/2/yr) = 21.08 + c1(Z − 90 − v)

A

+ c2(Z − 90 − v)2

A+ c3

(Z − 90 − v)3

A

+ c4(Z − 90 − v)

A(N − Z − 52)2, (5)

where c1 = −548.825021, c2 = −5.359139, c3 = 0.767379,c4 = −4.282220, v = 0 for the spontaneous fission of even-even nuclei and v = 2 for odd-A and odd-odd nuclei. Theseniority number v was introduced for taking the blockingeffect of unpaired nucleon on the transfer of many nucleon-pairs during the fission process. A variation of SF half-livescalculated by this formula [Eq. (5)] with increasing neutronnumber for different elements from Z = 102 to Z = 120are shown in Figs. 4 and 5 for comparison with the α

decay and SF half-lives predicted by our calculation and bySmolanczuk et al. [61,62], respectively. Moreover, in thiswork, spontaneous fission half-lives calculated in a dynamical

044603-4


Z=102

146 150 154 158 162

T1/

2(s

ec)

10-4

10-2

100

102

104

106

108

1010

(a)

Z-104

148 152 156 160 164

10-4

10-2

100

102

104

106

108

1010

(b)

Z=106

148 152 156 160 164 168

T1/

2(s

ec)

10-6

10-4

10-2

100

102

104

106

Z=108

154 158 162 166 17010-6

10-4

10-2

100

102

104

106

Z=110

N

152 158 164 170 176

T1/

2(s

ec)

10-10

10-8

10-6

10-4

10-2

100

102

104

106

Z=112

N

162 166 170 174 17810-10

10-8

10-6

10-4

10-2

100

102

104

106

(c) (d)

(e) (f)

Tα M3Y Q-M

Tsf Ph

Tsf SM

FIG. 4. (Color online) Plots for variation ofphenomenological [64,65] and microscopically[61,62] calculated fission half lives with neutronnumbers for (a) Z = 102, (b) Z = 104, (c) Z =106, (d) Z = 108, (e) Z = 110, (f) Z = 112.The corresponding α decay half-lives from thepresent calculation are also shown for comparison.Continuous line with solid circle (Tα M3Y Q-M)represents α decay half-lives predicted by thiswork using Q value from Muntian et al. [7,68,69].Continuous line with solid triangle (Tsf Ph) repre-sents spontaneous fission half-lives predicted byphenomenological calculation. Solid dark circle(Tsf SM) shows the spontaneous fission half-livespredicted by microscopic calculation.

approach using macroscopic-microscopic method (MMM) ofRefs. [61,62] are also used (Figs. 2–5) to find out the “island”where the predicted SHE could survive the fission.

IV. RESULTS AND DISCUSSION

In this paper, our main aim is to check whetherthe predicted SHN do really survive against the sponta-neous fission (SF). Gamow-Teller β-decay half-lives (Tβ inFigs. 2 and 3) obtained from a microscopic quasiparticlerandom phase approximation with single particle levels byMoller, Nix, and Kratz [63] has also been used in this workto check the possibility of β-decay of SHN. In earlier works[51,54,57–59], we were able to well reproduce the availableexperimentally measured α decay half-lives following thesemiclassical quantum tunneling method with double foldeddensity dependent effective M3Y interaction. In Ref. [58], Tα

of about 314 nuclei were predicted using Qα values fromMMM (QM ) [7,68,69] and modified liquid droplet model ofMyers-Swiatecki [67] to show the necessity of more accuratemass formula in determining the Qα-values with a good

accuracy at least correct upto 10 KeV for heaviest nuclei. Butthe spontaneous fission survivability of those nuclei was notchecked in our previous work. For this reason, although someSHN (Z = 102–120) are long-lived against α-decay [57,58],they may not live long if they have shorter SF half-lives. Forexample, at N = 162 the magnitudes of Tα using QM forthe elements No (Z = 102), Rf (Z = 104), Sg (Z = 106),Hs (Z = 108) are 3.06 × 109 s, 4.10 × 106 s, 1.16 × 104 s,28 s, respectively [58]. If SF half-lives (TSF) of such nuclei areabout the same order or relatively longer than their respectiveTα , only then can a significant fraction of nuclei survive fissionand decay by α emission. In such cases α-decay chain followedby SF of one of the product nuclei may be observed. Therefore,accurate estimation of TSF of SHN is essential to know whetherpredicted long-lived SHN against α emission do really existagainst SF.

Gupta and Burrows [78] summarized the measured groundstate values of spontaneous fission and α decay half-lives withQα for heaviest nuclei having mass number A = 266–294.These values were taken from experimental measurementscarried out at GSI, Germany, and JINR, Dubna. In Table I, acomparison between experimentally measured and calculated

044603-5


TABLE I. Comparisons between observed and theoretical (this work) α-decay half lives usingmeasured Qα . The experimental α-decay half lives (T EXP

1/2 ) are taken from Ref. [78] except thevalues with single (∗) and double (∗∗) asterisk symbols which are taken from Ref. [31] and Ref. [44]respectively.

Parent Exptl Q-value Half-lives (Exp) This work(� = 0) This work(� �= 0) �AZ Qex (MeV) T EXP

1/2 T DDM3Y1/2 [Qex] T DDM3Y

1/2 [Qex]

283112 9.704 ± 0.015 ∗6.9+6.9−2.3 s 4.67+0.49

−0.44 s277112 11.594 ± 0.055 0.69+0.69

−0.24 ms 93+29−23 µs 0.59+0.19

−0.14 ms 4272111 11.150 ± 0.035 3.8+1.4

−0.8 ms 1.31+0.27−0.22 ms

273110 11.37 ± 0.05 0.17+0.17−0.06 ms 75+21

−17 µs 0.13+0.04−0.03 ms 2

271110 10.899 ± 0.020 1.63+0.44−0.29 ms 930+110

−90 µs 1.15+0.14−0.12 ms 1

270110 11.20 ± 0.05 0.10+0.14−0.04 ms 0.083+0.024

−0.019 ms 0.10+0.03−0.02 ms 1

269110 11.58 ± 0.07 179+245−66 µs 28+13

−8 µs 88+36−26 µs 3

267110 12.28 ± 0.11 2.8+13.3−1.2 µs 1.1+0.8

−0.4 µs268109 10.486 ± 0.035 21+8

−5 ms 12.3+2.8−2.2 ms

266109 10.996 ± 0.025 1.7+1.8−1.6 ms 750+110

−90 µs 1.3+0.2−0.1 ms 2

270108 9.30+0.07−0.03 3.6+0.8

−1.4 s 1.36+0.30−0.51 s

∗∗270108 9.02 ± 0.03 ∗∗22 s 9.53+2.24−1.86 s

269108 9.315 ± 0.022 9.7+9.7−3.3 s 3.19+0.52

−0.43 s267108 9.978 ± 0.020 58+23

−14 ms 45.5+5.8−5.2 ms

266108 10.336 ± 0.020 2.3+1.3−0.6 ms 2.24+0.27

−0.25 ms267107 8.96 ± 0.30 17+14

−6 s 12+93−11 s

266106 8.88 ± 0.03 62+166−44 s 4.89+1.20

−0.93 s 16+4−3 s 3

half-lives using DDM3Y effective nucleon-nucleon (NN )interaction in a WKB framework is shown for 16 superheavynuclei. Most of these nuclei (12 out of 16) were not addressedin our earlier works [51,54,57–59]. In Fig. 1 the theoreticalQα-values from different mass formulas prescribed by MS[67], KUTY [66], Muntian et al. (MMM) [7,68,69] areemployed to calculate α-decay half-lives of the same 16SHN presented in Table I. The measured Qα-values are usedto calculate α decay half-lives (T DDM3Y

1/2 [Qex]) in Table I.Half-lives calculated in the present work agree reasonablywell with experimental data for most of the superheavy nuclei.Nuclei for which spin-parities of parent and daughter nuclei arenot known, zero angular momenta (� = 0) transfers are used.Since � = 0 gives minimum centrifugal barrier, probabilityof α-tunneling increases and consequently half-life decreases.But, for some nuclei like 277112, 273110, 269110, 266106, etc.,angular momenta carried by α particles may not be zero. Forthese nuclei, calculated half-lives are too small compared tomeasured values. In particular, for 277112 calculated value(∼0.093 ms) is less than the measured one (0.69 ms) byone order of magnitude. This discrepancy can be removedonly if one can assume nonzero orbital angular momentumtransfer in the α decay process. In our calculation, � = 4 givesthe better agreement of predicted half-life for 277112 nucleuswith measured value. It is therefore important to determinethe proper spin-parity of parent and daughter nuclei to enablecorrect centrifugal barrier for α decay half-life calculation.Although, higher �-values are shown for some nuclei for betteragreement with the experimental result, but more experiments

on the same nuclei (shown in Table I) with higher statisticsare needed for reconfirmation of the data and measured α

decay energies since reaction cross sections are of the order ofpicobarns (pb).

In this work, existing SF calculation with a microscopicapproach [61,62] have been used to find the region of long livedfission survived nuclei in the SHN region of the nuclear chart.In Figs. 2 and 3, comparisons between calculated Tα , Tβ , andTSF are shown only for even-Z elements with proton numberZ = 102–120 since Tα and TSF calculations of Refs. [61,62]are valid only for even-even nuclei. A few available observeddata for both Tα and TSF are also shown for most of theelements in the plots of Figs. 2 and 3. For Z = 104, thehighest value of Tα (∼4.1 × 106 s) according to calculationof this work using QM appears around N = 162 where muchsmaller value of TSF calculated in Ref. [61] (∼23 s) makes thisnucleus 266104162 unstable against SF. Therefore, if synthesisof this nucleus is possible in the present-day setup, substantialfraction of 266104162 would undergo SF within a few seconds.For Z = 102, no such calculation on SF is available inRefs. [61,62]. Since calculated TSF and QM values both arebased on MMM, α decay half-life calculations using DDM3Yeffective interaction with QM (Tα M3Y Q-M) is preferablychosen to compare with TSF. In addition to that, calculationwithin the same framework using QKUTY have also beenpresented in Figs. 1–3.

In the case of Sg (Z = 106) isotopes, DDM3Y with QM

predicts that the longest Tα (∼1.16 × 104 s ∼ 3.2 h) wouldbe at N = 162. It is comparable to TSF (3.5 h) of Ref. [61].

044603-6


Z=114

N150 156 162 168 174 180 186

T1/

2(s

ec)

10-17

10-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105

107

109

1011

1013

1015

(a)Z-116

N152 158 164 170 176 182 188

10-17

10-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105

107

109

1011

1013

1015

(b)

Z=118

N150 156 162 168 174 180 186

T1/

2(s

ec)

10-2110-1810-1510-1210-910-610-31001031061091012101510181021102410271030

Z=120

N152 158 164 170 176 182 188

10-2110-1810-1510-1210-910-610-31001031061091012101510181021102410271030

Tα M3Y Q-M

Tsf Ph

Tsf SM

(c) (d)

FIG. 5. (Color online) Same as Fig. 4 for (a)Z = 114, (b) Z = 116, (c) Z = 118, (d) Z =120.

Therefore, α decay channel is one of the dominant decay modeof this 268Sg162 nucleus. If it is produced in the laboratory,it may be observed only for few hours (lifetime ∼1.68 h)since both SF and α decay half-lives are small. It is to benoted that in the present work, lifetime of some SHN (eitherβ-stable or have large Tβ)are predicted by considering SFand α decay half lives only. Incidentally, 264No162, 266Rf162,268Sg162, and 270Hs162 are known to be either β-stable or havevery large Tβ [63]. Qα values using KUTY mass formula arenot always very much reliable since it does not reproduceall the observed Q-values with good accuracy. But, this massformula can be used to locate the region of possible existenceof long-lived SHN where Q-values from other mass formulaare not available. It may be pointed out that the use of QKUTY inour calculation shows reasonable agreement for several nucleiin Fig. 1.

As QM values are not available for more neutron richisotopes of Sg, using QKUTY in present calculation predictsTα ∼ 3.71 × 1015 s ∼1.18 × 108 y at N = 184. The SFhalf-life of 290Sg184 is less (TSF ∼ 4.07 × 1013 s ∼1.3 × 106 y)than Tα . Although Tβ of this nucleus is not specified but itis expected to be large in Ref. [63]. Hence, if synthesized,290Sg184 is expected to have a long enough lifetime (τ ∼1.27 × 106 y) to be observed in the laboratory. But it is stillsmaller than the age of earth ∼4.5 × 109 y by three orders

of magnitude. Calculated Tα using QM (28 s) for 270Hs162

is less than TSF (∼1.8 hr) and therefore α decay chain ofsuch nucleus is expected to be observed. This prediction fromour calculation is in good agreement with recently observed[44] doubly magic deformed nucleus 270Hs162. The presentcalculation using experimental Q-value gives Tα ∼ 9.53 s(Table I).

From Figs. 2 and 3 it is seen that the extra stability effect ofneutron shell at N = 162 almost disappears with the increaseof atomic number and Tα becomes of the order of millisecondto microsecond for the elements having Z � 110. On the otherhand, in more neutron rich side around N = 184 of elementsZ = 110, 112, 114 theoretical Tα using QKUTY in presentcalculation are of the order of 1010 s, 108 s, 106 s, respectively,which are much less than theoretical TSF of the order of 1012 s,1013 s, 1013 s, respectively. It must be noted that 296112184,298114184 are β-stable whereas 294110184 is predicted to havelarge Tβ in Ref. [63] due to its very small positive Qβ-value.β-stable nuclei and those with very large Tβ are not shown inFigs. 2 and 3.

Beyond the Z = 114 peak value of the Tα plot (Fig. 3)around N = 184 suddenly reduces showing a possible sig-nature of spherical proton shell at Z = 114. According tothe present calculation, 298114184 the so-called doubly magicspherical superheavy nucleus predicted by nuclear structure

044603-7


(b) Tαααα (MMM) vs. N for Z=102-120 (Even)

N

140 150 160 170 180 190

T1/

2(s

ec)

10-6

10-4

10-2

100

102

104

106

108

1010

116

118

120

(b)

(a) Tαααα (MMM) vs. N for Z=103-119 (Odd)

T1/

2(s

ec)

10-6

10-4

10-2

100

102

104

106

108

1010

103

105107

109111

113

115

117

119

102

104

106108

110

112114

FIG. 6. (Color online) Variation of α decay half-lives (Tα)predicted by present calculation with neutron number for (a) odd-Zand (b) even-Z elements from Z = 102–120. Each graphs show theabrupt reduction of Tα after N = 162 as the possible signature of shelleffect. Similar reductions of Tα after N = 184 are also shown by theelements for which Qα values are available from MMM calculation.

theory in the middle of 1960, has Tα values of the order of 106 sand 5 × 102 s using QKUTY and QM , respectively, which aremuch less than TSF ∼ 1013 s.

In Figs. 4 and 5, three curves describing spontaneous fissionand α-decay half-lives are shown in each graph except Z =102 for which calculation of spontaneous fission half-livesin Refs. [61,62], represented by Tsf SM in Figs. 4 and 5,are not available. For the isotopes of elements Z = 106, 108the values of Tsf SM become of the order of 1 ms nearN = 170. However, the values of Tsf SM are comparable tothe calculated α decay half-lives of this work (Tα M3Y Q-M)around N = 154 to 164 for Sg. In case of hassium iso-topes (Z = 108, N = 156–163), Tsf SM > Tα M3Y Q-Mreveals the fact that isotopes of Hs within this range mayundergo α decay with a longest half-life of the order offew seconds (∼10–30 s). This is in good agreement withthe recent experimental observation of α decay from thenucleus 270Hs162. But spontaneous fission half-lives calculatedusing Eq. (5) [64,65] fall rapidly around N = 160–170 for

elements having Z = 102–112. In Fig. 5, SF half-life valuesfor neutron rich isotopes of elements having Z = 114, 116,118, 120 are less than 10−15 s near N = 180. For Z = 114 only,this calculation matches with microscopic results for someisotopes having N = 160–170. This calculation contradictsthe following two facts: (i) Experimentally measured valueof Tα for 283112171 is ∼6.9 s (see Table I) whereas, SFhalf-life for this nucleus calculated by using Eq. (5) isextremely low (∼10−11 s) indicating immediate fissioning ofthe nucleus. (ii) Since TSF by Ren and Xu rapidly falls aroundN = 160–170 (Fig. 4) and N = 180 (Fig. 5) for Z = 102–112 and Z = 114–120, respectively, it could not explain thepredicted extra shell stability of SHN at N = 184 for heavierelements.

From the present calculation of Tα using QKUTY it seemsthat longer Tα might be observed for more neutron rich side(Z � 116, N > 190) due to possible existence of neutron shellclosure. On the contrary, from the trend of SF plots for Z =116, 118, 120 it appears that lowering of TSF might destroy theexistence of such neutron rich (N > 190) superheavy isotopesof elements Z = 116, 118, and 120. Hence, the presence oflong-lived SHN with neutron shell closure beyond N = 184may be ruled out. However, it is an important task to determinethe SF and α decay half-lives for N > 184 to confirm whetherthere is any possibility of neutron shell closure beyond N =184 for heaviest elements (Z � 116).

In two graphs of Fig. 6, using QM values in this calculation,variation of α decay with neutron number for both odd-Zand even-Z elements having Z = 102–120 are shown. Theplots of both graphs (a) and (b) of Fig. 6 clearly show peaksof Tα-values around N = 162 and N = 184 for all elementswith Z = 102–120, which possibly indicates the neutron shellclosure at N = 162 and 184. This is in good agreement withthe present-day knowledge of microscopic theory.

V. SUMMARY AND CONCLUSION

The natural existence of superheavy nuclei is limitedprimarily by spontaneous α decay and spontaneous fissionprocesses. A SHN in spite of having longer α decay half-livesmay undergo immediate spontaneous fission if the latter hasa low half-life. On the other hand, SF stable SHN may haveshorter (∼1 µs or less) α decay half-lives. In both the cases,such SHN may not be observed even if they are synthesized inthe present day laboratory setup. The main aim of this workis to find out the fission-survived long lived SHN. In fact,if SHN have high degree of stability against both α decayand SF, we would be able to observe them if produced inthe laboratory provided those SHN are not far away fromβ-stability line. We have calculated the α decay half-livesof SHN in quantum tunneling method with microscopic NN

potential using Q-values from different mass formulas andcompared them with the β-decay and SF half-lives to find thelong lived SHN.

The highlights of observations made in this work aresummarized as follows:

(i) Among all three mass formulas, Qα-values usedfrom MMM model (QM ) [7,68,69] in the present method,reproduces the observed data reasonably well (see Fig. 1), but

044603-8


nonavailability of Qα-values in more neutron rich side limitsits usage. Therefore, for a higher Z region, the mass formulaof KUTY, which extends up to Z = 130, has been considered.

(ii) Although 266Rf162 nucleus has relatively longer Tα half-life (∼4.1 × 106 s ∼47.5 d using QM ) but it is unstable againstSF with Tsf SM∼23 s only (Fig. 2).

(iii) The mass formula of KUTY predicts α decay lifetimeof the 290Sg184 nucleus to ∼108 y whereas Tsf SM∼106 ymakes the lifetime (∼106 y) of this nucleus very long butstill smaller than the age of the earth (∼4.5 × 109 y) by threeorders of magnitude. This nucleus is either β-stable or mighthave very large Tβ according to the calculations of Ref. [63].

(iv) The larger deviations between calculated and experi-mentally measured α decay half-lives are observed in case ofonly few nuclei such as 277112 which may be due to higherminimum orbital angular momenta carried away by α particlesfor spin-parity conservation. Inaccuracy in the measurementof Tα of 277112 nucleus due to very low count rate also maynot be ruled out.

(v) Using the formulation based on the liquid drop modelin Refs. [64,65] SF half-lives calculations have also beendone for higher Z elements in this work. Results shown inthe plots of Figs. 4 and 5 do not match with SF half-livescalculations in a microscopic approach of Refs. [61,62] forboth neutron rich and neutron deficient isotopes of heaviestelements with Z = 104–120. The half-life value using thisphenomenological prescription also contradicts the observedα-decay from 283112171 nucleus with measured Tα ∼ 6.9 s.

(vi) It is evident from the present Tα calculations that theeffect of the deformed neutron shell closure at N = 162 willbe insignificant beyond Z = 108 as Tα goes on decreasingwith increasing atomic number. For N = 162 isotopes ofelements having Z = 110, 112, values of TSF are 9.8 m, 0.63 s,respectively and Tα using QM are of the order of milliseconds(∼1 ms) and microseconds (∼93 µs) (see plots of Fig. 3),respectively, i.e., the values of Tα go on decreasing morerapidly than the corresponding SF half-lives with increasingatomic number.

(vii) Calculated Tα using QKUTY predicts almost β-stablelong lived SHN around 294110184, 296112184, 298114184 with Tα

of the order of ∼311 y, ∼3.10 y, ∼17 d, respectively, whichare much less than their TSF (∼4.48 × 104 y, ∼3.09 × 105

y, ∼4.38 × 105 y, respectively) values. Hence the dominantdecay mode of the above nuclei and their immediate neighborsis expected to have α decay mode. Tα value of 293110183 isabout 352 y which is slightly greater than that for 294110184

nucleus. The SF half-life of this nucleus is not found in

Refs. [61,62]. For 292108184 nucleus since Tα (∼9.6 × 104 yusing QKUTY) value is comparable to its TSF (∼3.2 × 104 y),this nucleus is one of the possible members of stability island.The exact values of Tβ of 293110183 and 292108184 nuclei arenot shown in Ref. [63] but predicted to be large.

(viii) Using QKUTY values in the present calculation showslonger Tα-values for neutron rich (N > 190) isotopes of Z =116, 118, and 120 indicating a possible neutron shell closurenext to N = 184 might occur. On the contrary, calculated SFhalf-lives (Fig. 3) of Refs. [61,62] show a trend of loweringof TSF (< 1 ms to 1 µs) for neutron rich isotopes of thoseelements which indicates the higher probability of SF of suchSHN in this region. However, more accurate determination offission barrier and their corresponding half-lives are essentialto predict long lived SHN in the region of very high atomicnumber.

(ix) It may be pointed out that the calculation of Tα is verysensitive to Qα-values, and none of the mass formula used herecover the entire mass range with extreme accuracy. Therefore,a better mass estimate covering the wide range of superheavymasses with a good accuracy is necessary.

In summary, we find that the possibility of existence ofSHN above Z = 114 with considerable life time is verylow. Although Z = 120, 124, 126 with N = 184 mightform spherical-doubly-magic nuclei and survive fission [79],they would undergo α decay within microseconds. A small“island/peninsula” might survive fission and β-decay butundergo α decay in the region Z = 106–108, N ∼ 160–164. Interestingly, in this region the β-stable SHN Z = 106,N = 162 has the highest α decay half-life ∼3.2 h (Fig. 2,using QM ) that is much greater than the recently discovereddeformed-doubly-magic SHN 270Hs (measured Tα ∼ 22 s).Thus a search for this long-lived SHN 268Sg162 can be pursued.Similarly, the nucleus with Z = 110, N = 183 appears to benear the center of a possible “magic island” (Z = 104–116,N ∼ 176–186) with α decay half-life ∼352 y (Fig. 3, usingQKUTY) which is greater than that of the doubly-magic SHNZ = 114, N = 184 (Tα ∼ 17 d). Since the SHN 290Sg184 hasTα and TSF values ∼108 y (Fig. 2, using QKUTY), and ∼106 y,respectively, it might have longer lifetime in comparisonto other superheavy nuclei. However, for both 293Ds183 and290Sg184 nuclei, β-decay might be another possible decay modewith large Tβ values. Only future experiments can confirmthis. Finally, the experimental investigations to detect the α-cascade can be pursued on 294110184, 293110183, 296112184, and298114184 nuclei which are expected to decay predominantlythrough α particle emission.

[1] W. D. Myers and W. J. Swiatecki, Nucl. Phys. 81, 1 (1966).[2] H. J. Krappe, J. R. Nix, and A. J. Sierk, Phys. Rev. C 20, 992

(1979).[3] Z. Patyk and A. Sobiczewski, Nucl. Phys. A533, 132

(1991).[4] R. Smolanczuk, A. Sobiczewski, Proceedings of the XV EPS

Conference on Nuclear Physics: Low Energy Nuclear Dynamics,St. Petersburg (Russia) 1995 (World Scientific, Singapore 1995),p. 313.

[5] W. D. Myers and W. J. Swiatecki, Nucl. Phys. A601, 141 (1996).

[6] R. R. Chasman and I. Ahmad, Phys. Lett. B392, 255 (1997).[7] I. Muntian, Z. Patyk, and A. Sobiczewski, Acta Phys. Pol. B 32,

691 (2001).[8] V. M. Strutinsky, Nucl. Phys. A122, 1 (1968).[9] S. Cwiok, J. Dobaczewski, P.-H. Heenen, P. Magierski, and

W. Nazarewicz, Nucl. Phys. A611, 211 (1996).[10] K. Rutz, M. Bender, T. Burvenich, T. Schilling, P.-G. Reinhard,

J. A. Maruhn, and W. Greiner, Phys. Rev. C 56, 238 (1997).[11] W. Greiner, J. Nucl. Radiochem. Sci. 3, 159 (2002).[12] Z. Ren, J. Nucl. Radiochem. Sci. 3, 195 (2002).

044603-9


[13] M. Bender, W. Nazarewicz, and P.-G. Reinhard, Phys. Lett.B515, 42 (2001).

[14] T. H. R. Skyrme, Phil. Mag. Nucl. Phys. 9, 615 (1958).[15] S. Cwiok , W. Nazarewicz, and P. H. Heenen, Phys. Rev. Lett.

83, 1108 (1999).[16] J. Decharge and D. Gogny, Phys. Rev. C 21, 1568 (1980).[17] J. F. Berger, M. Girod, and D. Gogny, Nucl. Phys. A428, 23c

(1984).[18] P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996).[19] G. A. Lalazissis, M. M. Sharma, P. Ring, and Y. K. Gambhir,

Nucl. Phys. A608, 202 (1996).[20] I. Muntian, Z. Patyk, and A. Sobiczewski, J. Nucl. Radiochem.

Sci. 3, 169 (2002).[21] Z. Patyk, A. Sobiczewski, P. Armbruster, and K. H. Schmidt,

Nucl. Phys. A491, 267 (1989).[22] Yu. A. Muzychka, Phys. Lett. B28, 539 (1969).[23] E. O. Fiset and J. R. Nix, Nucl. Phys. A193, 647 (1972).[24] V. M. Strutinski and Yu. A. Muzychka, Proceedings of the

International Conference on Heavy Ion Physics (JINR, Dubna,1966), Part 2, p. 51.

[25] G. N. Flerov and G. M. Ter-Akopian, Rep. Prog. Phys. 46, 817(1983).

[26] P. Armbruster, Acta Phys. Pol. B 34, 1825 (2003).[27] S. Hofmann et al., Z. Phys. A 350, 277 (1995).[28] S. Hofmann et al., Z. Phys. A 350, 281 (1995).[29] S. Hofmann et al., Z. Phys. A 354, 229 (1996).[30] S. Hofmann et al., Rep. Prog. Phys. 61, 639 (1998).[31] S. Hofmann et al., Eur. Phys. J. A 32, 251 (2007).[32] K. Morita et al., J. Phys. Soc. Jpn. 73, 2593 (2004).[33] K. Morita et al., J. Phys. Soc. Jpn. 76, 045001 (2007).[34] K. Morita et al., Eur. Phys. J. A 21, 257 (2004); T. N. Ginter

et al., Phys. Rev. C 67, 064609 (2003).[35] K. Morita et al., J. Phys. Soc. Jpn. 73, 1738 (2004).[36] K. Morita et al., J. Phys. Soc. Jpn. 76, 043201 (2007).[37] R. Eichler et al., Nucl. Phys. A787, 373 (2007).[38] Yu. A. Lazarev et al., Phys. Rev. C 54, 620 (1996).[39] Yu. Ts. Oganessian et al., Phys. Rev. Lett. 83, 3154

(1999).[40] Yu. Ts. Oganessian et al., Phys. Rev. C 74, 044602 (2006) , and

references therein.[41] Yu. Ts. Oganessian et al., Phys. Rev. C 69, 021601(R) (2004) ;

72, 034611 (2005).[42] Yu. Ts. Oganessian et al., Phys. Rev. C 70, 064609 (2004) ; 71,

029902(E) (2005).[43] Yu. Ts. Oganessian et al., Phys. Rev. C 76, 011601(R) (2007) ;

J. Phys. G 34, R165 (2007).[44] J. Dvorak et al., Phys. Rev. Lett. 97, 242501 (2006).[45] Z. Patyk and A. Sobiczewski, Phys. Lett. B256, 207 (1991).[46] A. Parkhomenko and A. Sobiczewski, Acta Phys. Pol. B 36,

3095 (2005).[47] S. G. Nilssson et al., Nucl. Phys. A115, 545 (1968).[48] S. G. Nilssson et al., Nucl. Phys. A131, 1 (1969).[49] J. Randrup, S. E. Larsson, P. Moller, A. Sobiczewski, and

A. Lukasiak, Phys. Scr. 10A, 60 (1974).

[50] D. N. Poenaru, W. Greiner, K. Depta, M. Ivascu, D. Mazilu, andA. Sandulescu, At. Data Nucl. Data Tables 34, 423 (1986).

[51] P. Roy Chowdhury, C. Samanta, and D. N. Basu, At. Data Nucl.Data Tables (in press).

[52] D. N. Poenaru, I-H. Plonski, and Walter Greiner, Phys. Rev. C74, 014312 (2006).

[53] D. N. Poenaru, I-H. Plonski, R. A. Gherghescu, and WalterGreiner, J. Phys. G: Nucl. Part. Phys. 32, 1223 (2006).

[54] C. Samanta, D. N. Basu, and P. Roy Chowdhury, J. Phys. Soc.Jpn. 76, 124201 (2007).

[55] D. S. Delion, R. J. Liotta, and R. Wyss, Phys. Rev. C 76, 044301(2007).

[56] H. F. Zhang and G. Royer, Phys. Rev. C 76, 047304 (2007).[57] P. R. Chowdhury, C. Samanta, and D. N. Basu, Phys. Rev. C 73,

014612 (2006).[58] C. Samanta, P. R. Chowdhury, and D. N. Basu, Nucl. Phys.

A789, 142 (2007).[59] P. Roy Chowdhury, D. N. Basu, and C. Samanta, Phys. Rev. C

75, 047306 (2007).[60] U. Mosel and W. Greiner, Z. Phys. 222, 261 (1969).[61] R. Smolanczuk, J. Skalski, and A. Sobiczewski, Phys. Rev. C

52, 1871 (1995).[62] R. Smolanczuk, Phys. Rev. C 56, 812 (1997).[63] P. Moller, J. R. Nix, and K.-L. Kratz, At. Data Nucl. Data Tables

66, 131 (1997).[64] C. Xu and Z. Ren, Phys. Rev. C 71, 014309 (2005).[65] Z. Ren and C. Xu, Nucl. Phys. A759, 64 (2005).[66] H. Koura, M. Uno, T. Tachibana, and M. Yamada, Nucl. Phys.

A674, 47 (2000).[67] W. D. Myers and W. J. Swiatecki, Lawrence Berkeley Laboratory

preprint LBL-36803, December (1994); Nucl. Phys. A601, 141(1996).

[68] I. Muntian, S. Hofmann, Z. Patyk, and A. Sobiczewski, ActaPhys. Pol. B 34, 2073 (2003).

[69] I. Muntian, Z. Patyk, and A. Sobiczewski, Phys. At. Nucl. 66,1015 (2003).

[70] I. Perlman and J. O. Rasmussen, Handbook of Physics (1957),Vol. XLII, p. 109.

[71] Flerov and Petrjak, Phys. Rev. 58, 89 (1940).[72] D. C. Hoffman, Nucl. Phys. A502, 21c (1989).[73] D. C. Hoffman, T. M. Hamilton, M. R. Lane, in “Spontaneous

Fission”, in Nuclear Decay Modes, edited by D. N. Poenaru(IOP, Bristol, 1996), Chap. 10, pp. 393–432.

[74] A. Baran, Z. Lojewski, and K. Sieja, Int. J. Mod. Phys. E 15,452 (2006).

[75] Niels Bohr and John Archibald Wheeler, Phys. Rev. 56, 426(1939).

[76] W. J. Swiatecki, Phys. Rev. 100, 937 (1955).[77] P. Moller, J. R. Nix, and W. J. Swiatecki, Nucl. Phys. A492, 349

(1989).[78] M. Gupta and Thomas W. Burrows, Nucl. Data Sheets 106, 251

(2005).[79] M. A. Stoyer, Nature (London) 442, 876 (2006), and references

therein.

044603-10

Documents

Search for long lived heaviest nuclei beyond the valley of stability